CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER
F. T. Howard Wake Forest University, Box 7388 Reynolda Station, Winston-Salem, NC 27109 (Submitted January 1993)
1. INTRODUCTION The Bernoulli polynomials of order k, for any integer k, may be defined by (see [10], p. 145):
2 x V = y^ #^ (wz,)*-". (i.i)
k k In particular, B^ \0) = B^ \ the Bernoulli number of order k, and BJp = Bn, the ordinary Bernoulli number. Note also that B^ = 0 for n > 0. The polynomials B^k\z) and the numbers B^ were first defined and studied by Niels Norlund in the 1920s; later they were the subject of many papers by L. Carlitz and others. For the past twenty-five years not much has been done with them, although recently the writer found an application for B^ involving congruences for Stirling numbers (see [8]). For the writer, the higher-order Bernoulli polynomials and numbers are still of interest, and they are worthy of further investigation. Apparently, not much is known about the divisibility properties of B^ for general k. Carlitz [2] proved that if/? is prime and kl k2 K k=alp +a2p +-- + arp {0
B%1)^P3 (mod/), (1.3) o B%=~-yBP* (mod/), (1.4) /? + l where Bp+l is the ordinary Bernoulli number. F. R. Olson [11] was able to extend (1.2) and (1.3) slightly by proving congruences modulo p6 and/?5, respectively. Carlitz [4] proved that B[p) is { p) integral (mod /?), p > 3, unless n = 0 (mod /? -1) and n = 0 or/? - 1 (mod /?), in which case pB n is integral. He also proved congruences for special cases of B^. The writer [8] examined the numbers B^ and proved that, for/? prime, /?>3, r odd, and p+l>r>5, P) r-4 &P =-'t—s{p,j)p^ (mod//), (1.5) 7=1 J + *
316 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER where s(p, j) is the Stirling number of the first kind. (The Stirling numbers are defined in section 2.) This enables us to extend (1.2), theoretically, to any modulus pr. Many other properties of B^ are worked out in [8], and applications are given that involve new congruences for the Stirling numbers. The purpose of the present paper is to examine the divisibility properties of B^ for arbitrary n and k. We are able to extend congruences (1.3) and (1.4), and we also generalize many of the results in [8] and [10]. A summary of the main results follows. 1. We prove that the Bernoulli polynomials have the following property: *&l*+\n)=(-ir*3&(-*+!») To the writer's knowledge, this is a new result. It is very helpful in proving congruences (1.6)- (1.9) below. 2. We extend (1.3) and (1.4) by proving, for p > 5:
* & 1 ) s - ^ ( P + 2)!/>a ( m o d / ) , (1.6)
2 &l&=^p (p + 2)\(p + l2bp+1) ( m o d / ) , (1.7)
! Bp ?4=^p\p + 4)K3p + 2)bp+3 ( m o d / ) , (1.8) where b„ is the Bernoulli number of the second kind, defined and studied by Jordan [9], pp. 265- 287 and by Carlitz [1]. The numbers bn are also defined in section 2 of this paper, and we show 1) in section 2 that B%~ = -{n-l)n\hn
3e Motivated by (1.6), we prove that if?? is odd and composite, n > 9, then Bj£l)^0 (mod«4). (1.9) 4. For k > 0, we define
Ak(P;n) = t ^ B^\ n\
and we prove that Ak(p; ri) is integral (mod/?); in fact, if/? does not divide k, then -^Ak(p; ri) is integral (mod/?). This improves results of Carlitz [2], [3],
So With Ak (p; ri) as defined above, we prove
4 ( p ; r ( p - l ) ) - ( - l ) r ( r ^ ) (mod/*), (1.10)
+ k Ak(p;r(p-l) + i) = ±(-iy-\r + k-l)(r k ^ (mQdp) (p>2) ( U 1 )
These congruences give us some insight into the highest power of/? (especially p = 2) dividing the denominator of B[n~k). This is discussed in sections 3 and 4.
1994] 317 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER
6. We prove the following recurrence formulas, which generalize results of Norlund [10], p. 150, for k = 0. For£>0,
n\ JzJ n + l-r r\ k (-\T B%k_f(n\ B& (» + *)! i?y){r + k)\-
These recurrences turn out to be helpful in proving (1.10), (1.11), and the fact that Ak(p;n) is integral (mod/?). Section 2 is a preliminary section that includes the definitions and known results that we need. In section 3 we examine B^ for arbitrary n and k, and we find new congruences, generat- ing functions and recurrences. In section 4 we look at B^n~l) in more detail, and we find some additional properties. Throughout the paper, the letter/? designates a prime number and the letter n denotes a non- negative integer. 2. PRELIMINARIES
We first note some special cases (see [4]). If n n x(x-l)--(x-n + l) = Yds(n,k)x\ (2.1) or by the generating function {log(l + x)}*=*!£s(H,*)4- tic n\ k) + k If k>0, then B^~ =(" k Y S(n + k, k), where S(n + k,k) is the Stirling number of the second kind, defined by x" = £ S(n, k)x(x -1) • • • (x - k +1), k=0 or by the generating function x k (e -\) =k\JjS(n,k)->'l», *J X Since the Stirling numbers are well known and have been extensively studied (see, e.g., [6], ch. 5; [8]; and [9], ch. 4), in this paper we will concentrate on B^k) for 0 < k < n. It follows from (1.1) that (see [10], p. 150): k B?\x+y) = ±{^x(x-\)...(x-j + \)Bi S/\y) = ± ^ (2.2) ±-B*\z) = nBM(z), (2.3) az B?\z 4-1) - £<*>(*) = nB&l\z). (2.4) 318 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER Norland [10], p. 145, proved B^iz) = ( l - | J B^(z) + (z-k)^_\(z), so that k) tfn-k) = * z £ JJC-^D + (t^LBtl . (2.5) k k Norland [10], p. 148, also proved Bik) =k<$£(-l)k-l-rs(k,k-r)^, (2.6) n-r which Is the basis for some of the results of Carlitz [3], such as (1.2)-(1.4) and the congruences p) for B^ . In (2.6), Bn_r is the ordinary Bernoulli number. Norlund [10], p. 147, proved the following integration formulas { x B :\x) = \ x (t-l)(t-2)-(t-n)dt, (2.7) B^ = -n\yt-l)-{t-n)dt, (2.8) which, when compared with (2.1), indicate the close relationship between the Stirling numbers and the higher-order Bernoulli numbers. Norlund [10], pp. 147 and 150, also gave the following generating functions: * = „ k y E»—x_ (29) {log(l + x)}* £> n-k n\ (l + x)log(l + x) £J n\ Jordan [9], pp. 265-87, defined and studied bn, the Bernoulli number of the second kind. The generating function is 7 - 7 ^ — =!>„*"• (2-11) log(l + x) „=0 Comparing (2.9), (2.10), and (2.11), we see that, for n * 1, -}-B^=n\bn=B^+nB<£\ (2.12) l-n The last equality also holds when n = 1. To the writer's knowledge, this relationship between hn and B^n~1^ has not been pointed out before. Jordan [9], p. 265, defined the polynomial ^ ( z ) , which has the generating function r^=t%w, (2.0) log(l + x) ^ 0 and he proved 1994] 319 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER ^ ^ - l + I n j = (-l)»«F^-z-l + i « j . (2.14) Carlitz [1] extended (2.13) in the logical way by defining /?^}(z): {log(l + *)}* to K)n\ k) Thus, fi^\z) is analogous to B^ (z), and $p(z) = n\Wn(z). Carlitz also proved the very useful result, fl+1\z-l) = B 5 ( ^ ) ( 1 ) = ^ + i ) ( 0 ) = ^ j J _ 5 ( ^ - i ) ; £(»>(!) = „l6n. (2.17) k + l-n 3. J5<*> for 0 < * £ it We first prove a theorem that is the basis for many of our later results. Theorem 3.1: For all nonnegative integers k, ^^+|»)=(-ir*^:^-z+i«j. (3.i) Proof: We use induction on k. The theorem is true for k = 0, since by (2.14) and (2.16) we have Assume (3.1) holds for a fixed & - 1 , i.e., 1 B$k-{z + £») = (-lr*- !^-* + \n) • Then, if » + & is even, ^"JUi^ + i") is an odd function ofz. By (2.3), this implies B$k{z + ±ri) is an even function ofz. That is, (3.1) holds for n + k even. If n + k is odd, then n +1 + k is even, and we apply the operator A to both sides of ^l^vH^HH" to get, by (2.4), J&[*44»=-I&-*-44» Letting y - z + ^-, we obtain, for n + & odd: 320 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER B^y + \n} = -B<$k{-y+\r^. This completes the proof. We note that Theorem 3.1 implies, for k > 0 and n > 0, #&(») = (-!)"***&. (3-2) n Now, since B^ \z + l) = «!*Fw(z), and since Jordan [9], p. 265, has shown d , ( z \ 1 ^ dz ™'{»-l)'(^^>-l.rV, it follows that ^ l J^1>(z) = (ii + l ) 2 ^ ^ r ) ( r - i r 1 + ^ 1 > ( l ) . (3.3) Equation (3.3) was also proved in [8], with different notation. Integrating (3.3) k times, using (2.3), we obtain I r+k+l (3.4) n+k +l l +i( r )Bi:t Ui)(z-iy. We now plug z = n + \ into (3.4). By (2.17) and (2.5), the first two terms in the last summation are *ffi(i)=(»+*+i)3&+*a, 1 (n + k + l)nB^ \l) = -k(rt + k + l)B^k, so by (3.2) we have, if n + k is odd: i ,, 1 r+ +i j(,, r (*-ix»+*+i)^=( iJt ^:( ? r - ^ (3.5) r=2 and if « + k is even, we have - i r+£+l (3.6) It is important to remember that (3.5) is valid when n + k is odd, and (3.6) is valid when n + k is even. We are now in a position to prove congruences (1.6)-(1.9). 1994] 321 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER Theorem 3.2: If j? is prime, p > 5, then B{ffi = —^(p + 2)\p2 (modp6). Proof: In (3.6), let n = p and k = 1. Then we have 5(/? r) +2 rtff> B - l ( p + 2)(p + 1)Y ' // (mod/). It is well known [5], pp. 218 and 229, that s(p,2j) = 0 (mod_/2/, ) Ij-l <- 7• .< 1- 0 ? - 3 ) | , so we have B%» =-±(p + 2)(p + l)s(p,l)p3 ^-±(p + 2)(p + l)(p-l)\p3 (modp6). This completes the proof. Theorem 3.2 extends Carlitz's congruence (1.3) and the work of Olson [11]. The motivation < p ) 2 for (1.3) was evidently the congruence B p ^ = 0 (mod/? ), which was proved by S. Wachs [12] in 1947. We will return to (3.5) later to prove congruences for B^2 and B^4. Next we prove two recurrence formulas that will be useful. Both formulas are given in [10], p. 150, for k = 0 only. Theorem 3.3: For k > 0, r = 0 „ - r x - r #•• Proof: In the first equation of (2.2), we replace n by n +1, we replace k by » +1 - k, and we let y = 0. We then subtract 5 ^ 1 - t ) from both sides and divide by x to obtain k) B^~ (»)-/&*-*> = ^ 4 - 1 ^ _ 1 ) ( x _ 2 ) , . . ( x _ . + D ^ y - y ) , ( 3 . 7 ) We now take the limit as x -» 0 of both sides of (3.7). The limit of the left side is lim —B^1^ (x) = (n + 1 ) ^ + 1 ^ } . Thus, we have and Theorem 3.3 follows by dividing both sides by (w + 1)! and letting r = n + l-j. This com- pletes the proof. Theorem 3.4: For k > 0, / i\«+fc n(n) n / \ o(r) (» + *)! •=%\rj(rz + k) 322 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER Proof: In the first equation of (2.2), replace n by n + k, replace k by #1, let y — 0, and let x — n. Theorem 3.4 now follows from (3.2), and the proof is complete. Now for k > 0, p prime, and [x] the greatest integer function, define (3.8) It was proved in [8] that A^{p\ ri) is integral (mod/?); we now show that Ak(p; ri) has that same property. We note that Ak(p; 0) = 1, by (2.9). Theorem 3.3 gives us n-\ p{nl{p-m-[rl{p-m 4 (p; ri) = 4 _ ! (p; w) - £ ZTV~Z At ( # r>- (3.9) r=0 n + l-r It was proved in [8] that if p%n +1 -r) then [nI(p-1)]-[r/(p-T)]>t. Therefore, we can use induction on k and on n in (3.9) to prove Ak(p;n) is integral (mod/?). In fact, it follows from (2.5) that - ^ Ak(p; ri) = - | A-ife *) + } A - i t e * - l ^ M H ^ M so if/? does not divide £, we see that -^Ak(p;ri) is integral (mod/?). Before putting this infor- mation together In a theorem, we make the following definitions. Let ap(n;k) denote the exponent of the highest power of p dividing the denominator of k B^~ ^ and let vp(ri) denote the exponent of the highest power of/? dividing n\ It Is well known that if 2 m n = nQ+nlP + n2p + --+nmp (Q n n then up(ri) = -^(n-^ ~ i m)- We can now state the following theorem. Theorem 3.5: Let/? be prime and let k > 0. Let n have base/? expansion (3.10) and let ap(n; k) and vp{ri) be as defined above. Then ap(n;k)< -vp(n) = v-\ p-l If pJ\(n - k) and p does not divide k, then ap(n;k)< -J- P-I Corollary: Suppose n has base/? expansion (3.10) and suppose WQ+WJH \-nm 1994] 323 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER Theorem 3.6: Let Ak(p; ri) be defined by (3.8). Then, for h > 0 andp prime, h + k Ak{p;h{p-Vj) = {-\) ^ k ^ (mod/;), (3.11) h + k Ak Proof: We will use equation (3.9). It was proved in [8] that we can have pw\(n + l-r) and - \ = w only when w = 0 orw = 1. Thus, we have, for 0 < t < p-1, A^p-Kp-Vt + ^^A^ip-hip-V + V-A^p-ih-lXp-V + t) r-i j -^———Ak(p;h(p-l)+i) (mod/?). In particular, for t = 0, we have ^ % - 1 ) ) - 4 I ( P ; A ( P - 1 ) ) - ^ ( P ; ( * - 1 ) ( P - 1 ) ) (mod/7). (3.13) In [8] it was proved that (3.11) is true for k = 0. Also, Ak(p; 0) = 1. Thus, we can use induction on k and on /i in (3.13) to prove that (3.11) is true for all k and h. To prove (3.12), we first note that Theorem 3.4 tells us that if n + k is odd, then 2 4 t e » ^ ) = Z^Vi)r+A4(p;^*)/'I(w+i)/(p-1)H(^)/(;'-1)]. Thus, 24taA(p-l) + l)s(A(p-l) + l-*)4foA(p-l)) and the proof is complete. For certain values of n, Theorem 3.6 gives us the exact value of ap(n; k). For example, suppose p = 2 and 2 m n = n0+nl2 + n22 + --+nm2 (0<^ < 1), 2 m n + k = t0 + tl2 + t22 +--+tm2 (0 By Theorem 3.6, we see that if kt < tt for all i, then a2(n; k) = n-u2(n) = n0 +nx + •*• +nm. (3.14) J In particular, if n - 2 , then a2 (/?; £) = 1 for all & * w; that is, if n is a power of 2, then 2, but not 4, divides the denominator of B^~k) for all k such that 0 324 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER Theorem 3.7: If p > 5, we have 2 B% =±p (p + 2)\(p + 12bp+l) (mod/), (3.15) 2 Bfi\^±p (p + 4)\(3p + 2)bp+3 (mod/), (3.16) where bn is the Bernoulli number of the second kind, defined by (2.11). In general, 2 B$Jk = 0 (mod/? ) {* = l,2,...,I(p-3)j. Proof: In (3.5), let n - p and let k - 2. Then we have 2 8% =^(p + 2)(p + l)s(p, \)p" + ^{p + 2)Bp^\\)p 3 2 = ±(p + 2)\p +±(p + 2)(Jp + l)\bp+1p (mod/), and (3.15) is proved. Now in (3.5) we let n = p and k = 4 to get 3(p + 5)Bpi\ = f f ^ V ^ O ) / / (mod/). r r=2 v By (2.17), Theorem 3.2, and (3.15), we see that 3 4 p Bp^(l) = 0 = p Bp^(l) ( m o d / ) . Thus, we have SJ&^(P+4)B&lW ( m o d / ) . (3.17) 6 By (2.5), (2.12), and Theorem 3.2, l) 2) B& (l) = -\(P + 1)B$ ^±(p + l)(p + 2)(p + 3)\bp+3 (mod/), (3.18) and we know (p + 3)\h3 is integral (mod/?) by (2.12). The proof of (3.16) now follows immedi- ately from (3.17) and (3.18). The last statement of Theorem 3.7 is clear from (3.5) and the proof is complete. We next derive another formula like (3.4). By (2.7) we have —B^iz) = nY$(n,r)zr~\ so B™ = nf-s{n,r)zr +B%\ dz ~ ~ r Integrating A: times, using (2.3), we get ^^c-) - (^ ijf (r^)_1-(« -^ -)^"+iO)^-}^^ • (3-i9) Equation (3.19) also follows directly from the second equality of (2.2). By (3.2) and (3.19) we have, for n + k odd, 1994] 325 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER -2^^= (2tl)|:(i^) ^ ^ ^ +±("^>^-y«y- (3-20) Carlitz [4] proved that B^ is integral (mod p), p>3, unless m = 0(modp-l) m\dm = 0 or p - 1 (mod p), in which case pB^ is integral. We note that by (3.20), with n + k = m and n = p, we can say: If m is odd, if p\m, and if p-1 does not divide m-l, then B^ = 0(mod/?2). 4. THE NUMBERS B ^ Because of their close relationship to the Bernoulli numbers of the second kind, that is, n 1 _1) B^ ~ ^ = (l-ri)n\bn (proved in section 2), the numbers J5^ deserve special consideration. We first note that, by (2.15) and (2.16), we have the generating function r 2 2 • = I # " (l + x){log(l + x)} ~0 " n\ If we integrate the right side of (2.8) we have, for n > 0, ^""1)=0-»)i-Lr^,'-X (4.1) which provides a way of computing 2^"_1) if a table of Stirling numbers is available. For example, Equation (4.1) was also given in [9], p. 267, as a formula for bn. Another useful formula is the following: If n is odd, then V sr=Q r + l Equation (4.2) follows from [9], p. 267, +l (n + l)\*¥n+2(z) = %— s(n + l,ry +(n + l ) \ h n + 2 , (4.3) Z^r + l x n where ^„(z) is defined by (2.13). If we plug z = n into (4.3) and use ¥n+2{n) = (-l) bn+2, which follows from (2.14), then (4.2) follows for odd n. We can now prove the following theorem. n l) 4 Theorem 4.1: lin is odd and composite, n > 9, then B^ +2 = 0 (mod n ). Proof: It was proved in [8] that if r >3 and n is odd and composite, n>9, then -^nr+ = 0 (mod??4). Thus, by (4.2), we have C 2 1 ) - ( " 2 2 ) { ^ ( " + 1'1)"2+l5(" + 1'2)"3} (mod«4)- (4-4> Now for n composite and n > 9 (see [6], p. 217), 326 [AUG. CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER $(n + \l) = -n\ = Q (modw2), $(n + l,2) = «! ! + - + - + • • • + - 1 = 0 (modw). J Also, we can easily see that if 3 \n9 then s(« + l,2) = 0 (mod3y+1) (j>2). Thus, Theorem 4.1 follows from (4.4), and the proof Is complete. For convenience, we again use the notation A1(p;n) = tAjL 5(»-D. n\ w-1) Because of (2.12), many properties of hn and^ follow from properties of B^\ Using results in [8], we can write down the following: 1 4(2; n) = 1 (mod8) (n * 1), (4.5) \-n —^-A^lrJ^i-iy-^Sp+Sr + l) (mod9), (4.6) 2 r - l — 4(3;2r + l) = (-l)r"1(4r3+3r2+l) (mod9) (r > 1). (4.7) 2r l) Congruence (4.5) gives us a2(n; 1), the exact power of 2 dividing the denominator of B^~ . J Using the notation of section 3, we have a2 («; 1) = n - v2 (n) - j = nQ + nx + • • • + nm - j , where 2 is the highest power of 2 dividing n-\ and w0,w1?..., nm are the digits in the base 2 expansion of n. Similarly, if n is not an odd integer congruent to 2 (mod 3), then (4.6) and (4.7) give a3(n91) = •u3(n)-j-- j . (4.8) J where 3 is the highest power of 3 dividing n-\ and nQ,nl9 ...,nm are the digits in the base 3 expansion of n. Ifn is an odd integer congruent to 2 (mod 3), we must replace the first "equals" symbol in (4.8) by "<" We know from section 3 that -^Ax{p\ n) is integral (modp) for any n ^ 1. Jordan [9], p. 267, proved {-l)n+\ >0 forn>0. Hence, we have (-l)nB^~l) >0(n> 1). In general, the sign of B^n~k^ is not known. It seems that the signs usually alternate when n-k>0, but there are exceptions. For example, B^ andBffl are both positive, B^ andB^ are both positive, B$ and B$ are both negative. ( n l) Norlund [10], p. 461, gave a table of values for B n ~ for n = 2,3,..., 12, and Jordan [9], p. n 1 266, listed hn for n = 0,1,2,..., 10. We give here the first fifteen values of B^ ~ ^ with numerators and denominators factored. 1994] 327 CONGRUENCES AND RECURRENCES FOR BERNOULLI NUMBERS OF HIGHER ORDER n- k > 0, but there are exceptions. For example, B^ and Bffi are both positive, B^ and i?^4) are both positive, B$ and B$ are both negative. { l) Norlund [10], p. 461, gave a table of values for B n"~ forn = 2,3,..., 12, and Jordan [9], p. _1) 266, listed bn for n = 0,1,2,..., 10. We give here the first fifteen values of J?^ with numerators and denominators factored. Table of the Numbers B ^ B^ = 0 5 ( 7 ) = 1494787 2 1 8 2-3 -5 / ? ( ! ) - J _ P(8) _ 2-73-167 ^ 2 ~ 2-3 ^ 9 ~ 5 2 9 # ( ) = _ 1 #( ) = 3-3250433 3 2 10 22-ll D ( 3 ) _ i 9 . p(10) _ 37-52-173 ^ 4 - 2-5 ^ 1 1 ~ 22 $ ( 4 ) - _ 9 ^C 1 1 ) - 11-541-4801-5273 5 1 2 22-3-5-7-13 o(5) _ 5-863 r>(12) _ ll3-2207-8329 6 " 22-3-7 1 3 " 2-5-7 £?(6) - 53-ll D(13) _ 13-132282840127 REFERENCES 1. L. Carlitz. "A Note on Bernoulli and Euler Polynomials of the Second Kind." ScriptaMath. 25 (1961):323-30. 2. L. Carlitz. "A Note on Bernoulli Numbers of Higher Order." ScriptaMath 22 (1955):217- 21. ( x) 3. L. Carlitz. "Some Properties of the Norlund Polynomial B n ." Math Nachr. 33 (1967): 297-311. 4. L. Carlitz. "Some Theorems on Bernoulli Numbers of Higher Order." Pacific J. Math. 2 (1952): 127-39. 5. L. Carlitz. "Some Congruences for Bernoulli Numbers of Higher Order." Quarterly J. Math 14 (1953): 112-16. 6. L. Comtet. Advanced Combinatorics. Dordrecht: Reidel, 1974. 7. F. T. Howard. "Congruences for the Stirling Numbers and Associated Stirling Numbers." ActaArith 55 (1990):29-41. 8. F. T. Howard. "Norlund's Number B^n)." In Applications of Fibonacci Numbers 5:355-66. Dordrecht: Kluwer, 1993. 9. C.Jordan. Calculus of Finite Differences. New York: Chelsea, 1965. 10. N. Norlund. Vorlesungen Uber Differenzenrechnung. New York: Chelsea, 1954. 11. F. R. Olson. "Arithmetical Properties of Bernoulli Numbers of Higher Order." Duke Math. J. 22(1955):641-53. 12. S. Wachs. "Sur une properiete arithmetique des nombres de Cauchy." Bull Set Math 71 (1947):219-32. AMS Classification Number: 11B68 *> *> • > 328 [AUG.